Problem: Rewrite as a simplified fraction. ${0.\overline{90} = {?}}$
Answer: Let $x$ equal the decimal. Set up two equations such that the digits after the decimal point are identical. ${\begin{align*}100x &= 90.909...\\ 1x &= 0.909...\end{align*}}$ Subtracting the two equations, we have: ${99x = 90}$ ${x = \dfrac{90}{99}} $ ${x = \dfrac{10}{11}} $